Properties of Fractals
(What is a fractal anyway?)
Inﬁnity by ﬂickr user azarius
The Sierpinski Triangle
Waclaw Sierpinski, a Polish mathematician, developed another fractal known
as the Sierpinski Triangle. This fractal also starts with an equilateral triangle.
To draw the fractal, you find the midpoint of each side of the original triangle,
and then draw three segments joining the midpoints. There are now four
triangles inside the original triangle. The middle triangle is not shaded, and the
process is continued with the other three shaded triangles, as shown in the
The Koch Snowflake
This fractal -- the Koch Snowflake -- was developed by in 1904 by Helge von
Koch, a Swedish mathematician. The fractal is started by drawing an
Each side of the triangle is trisected, and the middle section forms the base of
a new equilateral triangle outside the original one.
The process is then continued. The diagram below shows three generations of
the Koch Snowflake.
1st iteration 2nd iteration
Draw a Fractal
Use pencil and paper (metric graph paper if possible) to draw the fractal
• Draw a square with 8-cm sides in the middle of the paper.
• Position the paper horizontally (in landscape format). Extend the
diagram to the left and right by drawing a square on each side of the
original square -- touching the original square. The sides of the new
squares should be half as long as the side lengths of the original square.
• Repeat the previous step three times. Your fractal should now have
five generations, including the original square.
Question: Will the fractal ever be too large for this page? Explain.
The Rectangle ...
Draw a rectangle that measures 12 cm by 8 cm, and shade the inside of the
rectangle. Construct the midpoints of each side of the rectangle, and then draw
a quadrilateral by joining these points. Shade the quadrilateral white. Now
continue the process by finding the midpoints of the quadrilateral, drawing the
rectangle, and shading it the same colour as the first rectangle. Draw six
generations. (The initial rectangle is the first generation.)
The Square ...
Create a fractal that begins with a large square 20 cm on each side. Each
pattern requires that the square be divided into four equally sized squares, that
the bottom-left square be shaded, and the process continues in the upper-right
square. Repeat the process four times.